Logic and sets · Set algebra · Symmetric difference · Venn regions
Let \(A\), \(B\) and \(C\) be finite sets whose sizes satisfy
\[ |A \triangle C| + |B \triangle C| = |A \triangle B|. \]
Prove that \(C\) is then trapped between the intersection and the union of the other two:
\[ A \cap B \subseteq C \subseteq A \cup B. \]
Here \(X \triangle Y = (X \setminus Y) \cup (Y \setminus X)\) denotes the symmetric difference of \(X\) and \(Y\): the elements belonging to exactly one of the two sets.
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Serbian Municipal Competition 2016, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source